On the tails of Pitman–Yor random probability measures: Transport maps and stick-breaking constructions

Abstract While random probability measures have a long tradition in probability and statistics, little is known about their tails. The few available results are derived using subordinators, and therefore only apply to measures that can be represented as normalized subordinators, such as the Dirichlet process. Our work breaks this barrier, by exploiting the stick-breaking representation to construct a new family of transport maps that preserve the decay of tails. Drawing on recent developments on regular variation and on subordinator theory, the new family of maps allows us to establish that the right tail of a Pitman–Yor process is heavy-tailed if the centering distribution is itself heavy-tailed; the Dirichlet process is the only member of this class that fails to obey this convenient property. Asymptotic envelopes for the tails of the Pitman–Yor processes are also derived. Finally, we discuss some consequences of the main results, including aspects related to the posterior distribution.